# Function defined over all a set (NOTATION)

I want to write down symbolically a function that is defined over a set (all the set, no just an element of it) to the natural numbers.

The function is something like this: For example $X\subset \mathbb{Z}^+$ is a set with finite number of elements. My function $R(\cdot)$ take the set $X$ and draw an element from it at random. For example $R(\{1,2,3\})=2$

I want to use the notation $$f : A \to B$$ But i don't know what to put in the place of $A$.

• so you want a symbolic notation for a function which maps the entirety of A To B? – Vaas Dec 3 '17 at 1:30
• The power set of $\mathbb{Z}$? Or $\{A\}$? – Nightgap Dec 3 '17 at 1:30
• "My function $R(\cdot)$ take the set $X$ and draws an element from it at random." Are you saying that at times $R(\{1,2,3\})=2$ and at other times $R(\{1,2,3\})=1$? If so, then this is not a function because it is not well-defined. What you may be looking for is instead a random variable. – JMoravitz Dec 3 '17 at 1:44
• Oh, yes. You are right. I might use "application" instead of "function" – Gabriel Sandoval Dec 3 '17 at 1:46

Use power set:

$R: \mathcal{P}(A) \to A$ such that $R(X) \in X$.

Another notation to power set is $2^A$.

If you're thinking some function that each time associate each subset to a element, it's something like

$R: \mathbb{Z} \times \mathcal{P}(A) \to A$ such that $R(X) \in X$.

Where $\mathbb{Z}$ or $\mathbb{R}$ can simulate time/events.

If you're thinking a function that take JUST ONE value: associate JUST $A$ to an element (and not all subsets of $A$ to an element), it is

$R: \{A\} \to A$

• It's a choice function. – Thadeu Henrique Costa Dec 3 '17 at 1:36
• is $R: \{ A\} \to A$ equivalent to $R: \mathcal{P}(A) \to A$? – Gabriel Sandoval Dec 3 '17 at 1:41
• $\{ A\}$ is a set with just one element: the set $A$. $\mathcal{P}(A)$ is the set of ALL subsets of $A$. For example, if $A = \{0,1,2\}$, them $\mathcal{P}(A) = \{\varnothing, \{0\}, \{1\} , \{2\}, \{0,1\}, \{0.2\}, \{1,2\}, A \}$. Note: $card (\mathcal{P}(A)) = 2^{card(A)}$, this is one reason to use $2^A$. – Thadeu Henrique Costa Dec 3 '17 at 4:29
• Another reason is that a subset of $A$ can be modeled by a function: $\chi_X: A \to {0,1}$ such that $\chi_X(a) = 0$ if $a \in X$ and $\chi_X(a)=1$ if $a \in A\setminus X$. In a set-theoretic construction of $\mathbb{N}$, $2 = \{0,1\}$. Then, the power-set of A is in one-to-one correspondence with the set of function from $A$ to $2$, i.e., $2^A$. – Thadeu Henrique Costa Dec 3 '17 at 4:34

$f:2^A\rightarrow B$ defines a function that maps every subset of $A$ to an element of $B$.

• its either this or he means f is surjective. – Vaas Dec 3 '17 at 1:31